|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > tpcomb | Structured version Visualization version GIF version | ||
| Description: Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.) | 
| Ref | Expression | 
|---|---|
| tpcomb | ⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tpcoma 4749 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝐶, 𝐵, 𝐴} | |
| 2 | tprot 4748 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
| 3 | tprot 4748 | . 2 ⊢ {𝐴, 𝐶, 𝐵} = {𝐶, 𝐵, 𝐴} | |
| 4 | 1, 2, 3 | 3eqtr4i 2774 | 1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 {ctp 4629 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 df-tp 4630 | 
| This theorem is referenced by: f13dfv 7295 frgr3v 30295 signswch 34577 signstfvcl 34589 dvh4dimN 41450 | 
| Copyright terms: Public domain | W3C validator |